I I £ CHAP. I] PERFECT, MULTIPLY PEKFECT, AND AMICABLE NUMBERS. 5 fu ri. TMbit ben Korrah,10 in a manuscript composed the last half of the 5* ninth century, attributed to Pythagoras and his school the employment of ^ perfect and amicable numbers in illustration of their philosophy. Let ^ s==i-|_2+...+2n. Then (prop. 5), 2 Vis a perfect number if s is a prime; 1^ 2np is abundant if p is a prune <s, deficient if p is a prime >s, and the excess or deficiency of the sum of all the divisors over the number equals rj the difference of s and p. Let (prop. 6) p' and p" be distinct primes >2; " the sum of the divisors <N of N=p'p"2n is a = (2n+1 -1) (1 +p'+p'f) + (2^- l)p'p'. Hence N is abundant or deficient according as p//)-py/>0 or <0. Hrotsvitha,11 a nun in Saxony, in the second half of the tenth century, mentioned the perfect numbers 6, 28, 496, 8128. Abraham Ibn Ezralla (fl!67), in his commentary to the Pentateuch, Ex. 3, 15, stated that there is only one perfect number between any two successive powers of 10. Rabbr Josef b. Jehuda Ankin11&, at the end of the twelfth century, recommended the study of perfect numbers in the program of education laid out in his book "Healing of Souls." Jordanus Nemorarius12 (f!236) stated (in Book VII, props. 55, 56) that every multiple of a perfect or abundant number is abundant, and every divisor of a perfect number is deficient. He attempted to prove (VII, 57) the erroneous statement that all abundant numbers are even. Leonardo Pisano, or Fibonacci, cited in his Liber Abbaci13 of 1202, revised about 1228, the perfect numbers i 22(22-l)-6, } 23(23-l)=28, fc 25(25-l)=496, excluding the exponent 4 since 24 —1 is not prime. He stated that by proceeding so, you can find an infinitude of perfect numbers. "Manuscript 952, 2, Suppl. Arabe, Bibliotheque imp6riale, Paris. Textual trans!., except of the proofs which arc given in modern algebraic notation as foot-notes [as numbers were represented by line, in the manuscript], by Franz Woepcke, Journal Asiatique, (4), 20, 1852, 420-9. "See Ch. Magnin, Th&itre de Hrotsvitha, Paris, 1845. llflMikrooth Gcdoloth, Warsaw, 1874 ("Large Bible" in Hebrew). Samuel Ben Saadias Ibn Motot, a Spaniard, wrote in 1370 a commentary on Ibn Ezra's commentary, Perush ai Perush Ibn Ezra, Venice, 1554, p. 19, noting the perfect numbers 6, 28, 496, 8128, and citing Euclid's rule. Steinschneider, in his book on Ibn Ezra, Abh. Geschichte Math. • Wiss., 1880, p. 92, stated that Ibn Ezra gave a rule for finding all perfect numbers. As this rule is not given in the Mikrooth Gedoloth of 1874, Mr. Ginsburg of Columbia University infers the existence of a fuller version of Ibn Ezra's commentary. "^Quoted by Gudeman, Das Jiidische Untcrrichtswesen wtihrend der Spanish Arabischen Periode, Wien, 1873. 12In hoc opere contonta. Arithmetica decem libris dcmonstrata .... Epitome I libros arithmetics diui Seuerini Boetij . . . , Paris, 1496, 1503, etc. It contains Jordanus' "Elementa arithmetica decem libris, demonstrationibus Jacobi Fabri Stapulensis," and " Jacobi Fabri Stapulensis epitome in duos libros arithmeticos diui Seuerini Boetij." 13I1 Liber Abbaci di Leonardo Pisano. Roma, 1857, p. 283 (Scritti, vol. 1).